Asymptotic boundary layer method for unstable trajectories: Semiclassics for individual scar wavefunctions

TL;DR

This paper extends the asymptotic boundary layer method to describe individual scar wavefunctions localized around unstable periodic orbits in chaotic quantum systems, providing a semiclassical approach to analyze their structure.

Contribution

The authors develop a formalism that generalizes the ABL method for unstable trajectories, applicable to systems with various potentials and magnetic fields, and validate it against numerical solutions.

Findings

ABL formalism accurately predicts scar wavefunctions

Separable wave functions dominate scar contributions

Good agreement with numerical solutions in specific resonator models

Abstract

We extend the asymptotic boundary layer (ABL) method, originally developed for stable resonator modes, to the description of individual wavefunctions localized around unstable periodic orbits. The formalism applies to the description of scar states in fully or partially chaotic quantum systems, and also allows for the presence of smooth and sharp potentials, as well as magnetic fields. We argue that the separatrix wave function provides the largest contribution to the scars on a single wave function. This agrees with earlier results on the wave-function asymptotics and on the quantization condition of the scar states. Predictions of the ABL formalism are compared with the exact numerical solution for a strip resonator with a parabolic confinement potential and a magnetic field.